Optimal Power for Testing Potential Cointegrating Vectors With Known Parameters for Nonstationarity

Abstract

Theory often specifies a particular cointegrating vector among integrated variables, and testing for a unit root in the known cointegrating vector is often required. Although it is common to simply use a univariate test for a unit root for this test, it is known that this does not take into account all available information. We show here that in such testing situations, a family of tests with optimality properties exists. We use this to characterize the extent of the loss in power from using popular methods, as well as to derive a test that works well in practice. We also characterize the extent of the losses of not imposing the cointegrating vector in the testing procedure. We apply various tests to the hypothesis positing that price forecasts from the Livingston data survey are cointegrated with prices, and find that although most tests fail to reject the presence of a unit root in forecast errors, the tests presented here strongly reject this (implausible) hypothesis.

Combining non-cointegration tests

“…See for instance Pesavento (2004Pesavento ( , 2007 or Elliott et al (2005) for DGP(A), Swensen (2006) for (B), or Engle and Granger (1987), Haug (1996) and Gregory et al (2004) for (C).…”

“…This choice is difficult because, as discussed in e.g. Elliott et al (2005), there exists no uniformly most powerful test, even asymptotically. Often one test rejects the null hypothesis whereas another test does not, making it unclear how to interpret test outcomes then.…”

Residuals‐based tests for cointegration with generalized least‐squares detrended data

“…This is the same DGP adopted by, among others, Elliott, Jansson and Pesavento (2005) to analyze optimal tests with a known cointegrating vector, Pesavento (2004Pesavento ( , 2007 to analyze the local asymptotic power functions of various tests for cointegration, and Zivot (2000) to analyze the power of single equation ECM tests for cointegration when the cointegrating vector is known. It permits non-weakly exogenous regressors.…”

Efficient Tests under a Weak Convergence Assumption

“…This second step is important, because in most applications, one would not want to assume that the canonical model is necessarily correct. Examples of this approach include the univariate unit root tests by Elliott, Rothenberg, and Stock (1996) (subsequently abbreviated ERS), Elliott (1999), Müller and Elliott (2003), and Müller (2009); the unit root test with stationary covariates by Elliott and Jansson (2003); the test of the null hypothesis of no cointegration with known cointegrating vector by Elliott, Jansson, and Pesavento (2005); the test of the null hypothesis of cointegration by Jansson (2005); the parameter stability tests by Nyblom (1989), Andrews and Ploberger (1994), and Elliott and Müller (2006); the tests about regression coefficients 1 A previous version of this paper was circulated under the title "An Alternative Sense of Asymptotic Efficiency." The author would like to thank Whitney Newey, three anonymous referees, Mark Watson, seminar participants at Berkeley, Brown, Harvard/MIT, Penn State, and Stanford, and conference participants at the NBER Summer Institute 2008 and the ESEM Summer Meeting 2008 for helpful comments and discussions, as well as Jia Li for excellent research assistance.…”